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300

The smallest squares containing k 300's :
63001 = 251²,
3000300625 = 54775²,
109630030030096 = 10470436².

300² + 301² + 302² + ... + 312² = 313² + 314² + 315² + ... + 324².

(1 + 2 + 3 + 4)(5)(6 + 7 + 8 + 9)(10 + 11 + 12 + 13 + 14) = 300²,
(1)(2 + 3)(4 + 5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14) = 300²,
(1)(2 + 3 + 4)(5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14) = 300²,
(1 + 2 + ... + 4)(5 + 6 + 7)(8 + 9 + ... + 32) = 300².

300² = 90000 appears in the decimal expression of π
  π = 3.14159•••90000••• (from the 49054th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

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301

The smallest squares containing k 301's :
301401 = 549²,
47301030144 = 217488²,
130187301301225 = 11409965².

301² = 90601, a zigzag and reversible square (10609 = 103²).

Cubic Polynomial (X + 192²)(X + 252²)(X + 301²) = X³ + 437²X² + 106932²X + 14563584².

29^k + 109^k + 301^k + 461^k are squares for k = 1,2,3 (30², 562², 11250²).

3-by-3 magic squares consisting of different squares with constant 301²:

301² + 302² + 303² + 304² + 305² + ... + 326² = 1599²,
301² + 302² + 303² + 304² + 305² + ... + 925² = 15975²,
301² + 302² + 303² + 304² + 305² + ... + 1574² = 35945².

301² = 90601, 9 + 0 + 6 + 0 + 1 = 4².

301² = 90601 appears in the decimal expression of e:
  e = 2.71828•••90601••• (from the 33432ndth digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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302

The smallest squares containing k 302's :
3025 = 55²,
3023020324 = 54982²,
153430213023025 = 12386695².

302² = 91204, a zigzag square with different digits.

302² + 303² + 304² + 305² + 306² + ... + 855² = 14127²,
302² + 303² + 304² + 305² + 306² + ... + 94550² = 16785532².

302² = 91204, 9 + 1 + 2 + 0 + 4 = 4²,
302² = 91204, 9 + 12 + 0 + 4 = 5².

302² = 91204 appears in the decimal expressions of π and e:
  π = 3.14159•••91204••• (from the 66475th digit),
  e = 2.71828•••91204••• (from the 44293rd digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised June 26, 2006
by Yoshio Mimura, Kobe, Japan

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303

The smallest squares containing k 303's :
130321 = 361²,
3033035329 = 55073²,
533303830330384 = 23093372².

303² = 91809, a zigzag square.

303²± 2 are primes (the 9th case).

303² = 10³ + 33³ + 38³.

Komachi Fraction : 303² = 7436529 / 81.

Komachi equation: 303² = 9 + 8 * 765 / 4 * 3 * 2 * 10.

3-by-3 magic squares consisting of different squares with constant 303²:

303² = 91809, 9 + 18 + 0 + 9 = 6²,
303² = 91809, 91 + 809 = 30².

303² = 91809 appears in the decimal expressions of π and e:
  π = 3.14159•••91809••• (from the 14617th digit),
  e = 2.71828•••91809••• (from the 105000th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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304

The smallest squares containing k 304's :
2304 = 48²,
304572304 = 17452²,
304304483044 = 551638².

The squares which begin with 304 and end in 304 are
304572304 = 17452²,   30433500304 = 174452²,   30467004304 = 174548²,
304099308304 = 551452²,   304205196304 = 551548²,...

304² = 92416, a zigzag square with different digits.

304²± 3 are primes.

81^k + 136^k + 304^k + 704^k are squares for k = 1,2,3 (35², 783², 19495²).

304² + 305² + 306² + 307² + 308² + ... + 327² = 1546²,
304² + 305² + 306² + 307² + 308² + ... + 655² = 9196².

304² = 92416, 9 + 24 + 16 = 7²,
304² = 92416, 9 + 241 + 6 = 16².

304² = 92416 appears in the decimal expressions of π and e:
  π = 3.14159•••92416••• (from the 41626th digit),
  e = 2.71828•••92416••• (from the 43455th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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305

The smallest squares containing k 305's :
173056 = 416²,
305305729 = 17473²,
1453053053058489 = 38118933².

305² = 93025, a square with different digits.

305² = 93025 with 9 = 3² and 3025 = 55².

A + B, A + C, A + D, B + C, B + D, and C + D are squares for A = 305, B = 1376, C = 2720, D = 5024).

98^k + 212^k + 305^k + 346^k are squares for k = 1,2,3 (31², 517², 8959²).
125^k + 241^k + 305^k + 485^k are squares for k = 1,2,3 (34², 634², 12586²).

3-by-3 magic squares consisting of different squares with constant 305²:

305² = 93025, 9 + 30 + 25 = 8²,
305² = 93025, 93 + 0 + 2 + 5 = 10².

305² = 93025 appears in the decimal expression of e:
  e = 2.71828•••93025••• (from the 44493rd digit).

3007² = 112² + 113² + 114² + 115² + 116² + ... + 305²,
3069² = 64² + 65² + 66² + 67² + 68² + ... + 305².

Page of Squares : First Upload November 22, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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306

The smallest squares containing k 306's :
30625 = 175²,
3063069025 = 55345²,
230630663067841 = 15186529².

306² = 93636, a zigzag square with 3 kinds of digits.

306² = 93636, 9 * 36 - 3 * 6 = 936 / 3 - 6 = 306.

306² = 102² + 204² + 204² : 402² + 402² + 201² = 603².

306² = (1² + 2)(2² + 2)(7² + 2)(10² + 2) = (2² + 2)(5² + 2)(24² + 2)
= (3² + 9)(5² + 9)(12² + 9) = (4² + 2)(7² + 2)(10² + 2).

240^k + 306^k + 313^k + 366^k are squares for k = 1,2,3 (35², 619², 11053²).
3570^k + 20706^k + 21522^k + 47838^k are squares for k = 1,2,3 (306², 56508², 11329956²).
8466^k + 14178^k + 27438^k + 43554^k are squares for k = 1,2,3 (306², 54060², 10331172²).

306² = 93636 with 9 = 3² and 36 = 6².

306² = 93636, 9 + 3 + 63 + 6 = 9²,
306² = 93636, 9 + 36 + 36 = 9²,
306² = 93636, 93 + 636 = 27².

306² + 307² + 308² + 309² + 310² + ... + 12604² = 817005².

306² = 93636 appears in the decimal expressions of π and e:
  π = 3.14159•••93636••• (from the 14211st digit),
  e = 2.71828•••93636••• (from the 12444th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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307

The smallest squares containing k 307's :
1503076 = 1226²,
307230784 = 17528²,
839430730730761 = 28972931².

3-by-3 magic squares consisting of different squares with constant 307²:

307² = 94249, 9 + 4 + 2 + 49 = 8²,
307² = 94249, 9 + 42 + 4 + 9 = 8²,
307² = 94249, 9 + 42 + 49 = 10².

307² = 94249, a palindromic square with 3 kinds of digits.

307² = 94249 appears in the decimal expressions of π and e:
  π = 3.14159•••94249••• (from the 53998th digit),
  e = 2.71828•••94249••• (from the 74007th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised January 27, 2009
by Yoshio Mimura, Kobe, Japan

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308

The smallest squares containing k 308's :
308025 = 555²,
30843086884 = 175622²,
2530820308093081 = 50307259².

1 / 308 = 0.00324..., 324 = 18².

308² = 94864, 9 + 48 + 64 = 11²,
308² = 94864, 94² + 8² + 64² = 114².

7315^k + 23177^k + 28259^k + 36113^k are squares for k = 1,2,3 (308², 51898², 9083228²).

3128² = 20² + 21² + 22² + 23² + 24² + 25² + ... + 308².

308² = 94864 appears in the decimal expressions of π and e:
  π = 3.14159•••94864••• (from the 102495th digit),
  e = 2.71828•••94864••• (from the 108719th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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309

The smallest squares containing k 309's :
30976 = 176²,
13093309476 = 114426²,
281309309309284 = 16772278².

309² = 95481, a square with different digits.

309²± 2 are primes (the 10th case).

3-by-3 magic squares consisting of different squares with constant 309²:

309² = 95481, 9 + 54 + 81 = 12²,
309² = 95481, 95 + 48 + 1 = 12²,
309² = 95481, 95 + 481 = 24².

309² + 310² + 311² + 312² + 313² + ... + 837² = 13639².

(1³ + 2³ + ... + 59³)(60³ + 61³ + ... + 155³)(156³ + 157³ + ... + 309³) = 981042678000².

309² = 103² + 206² + 206² : 602² + 602² + 301² = 903².

309² = 95481 appears in the decimal expressions of π and e:
  π = 3.14159•••95481••• (from the 40335th digit),
  e = 2.71828•••95481••• (from the 19170th digit).

Page of Squares : First Upload November 22, 2004 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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